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%I #27 Jan 26 2020 05:10:04
%S 1,2,10,140,5544,622908,198846076,180473355920,465904151957920,
%T 3422048076740462480,71525763221287897903500,
%U 4254840960508487045451825000,720428791920558617462950575000000,347230535542092373572967034254050000000
%N Number of 2n X 2n half-turn symmetric alternating-sign matrices (HTSASM's).
%H Seiichi Manyama, <a href="/A059475/b059475.txt">Table of n, a(n) for n = 0..66</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H J. de Gier, <a href="https://arxiv.org/abs/math/0211285">Loops, matchings and alternating-sign matrices</a>, arXiv:math/0211285 [math.CO], 2002-2003.
%H Heuer, Dylan, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, and Corey Vorland. "Chained permutations and alternating sign matrices - Inspired by three-person chess." Discrete Mathematics 340, no. 12 (2017): 2732-2752. Also <a href="http://arxiv.org/abs/1611.03387">arXiv:1611.03387</a>.
%H G. Kuperberg, <a href="https://arxiv.org/abs/math/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv:math/0008184 [math.CO], 2001.
%F a(n) = A005130(n)*A006366(n).
%F a(n) = A049503(n)*Product_{k=0..n-1} (3*k+2)/(3*k+1). - _Seiichi Manyama_, Jul 29 2018
%F a(n) ~ exp(1/18) * Gamma(1/3)^(2/3) * n^(1/18) * 3^(3*n^2 + 1/9) / (A^(2/3) * Pi^(1/3) * 2^(4*n^2 + 1/6)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jan 26 2020
%t a[n_] := Product[(3k+1)(3k+2)(3k)!^2/(n+k)!^2, {k, 0, n-1}];
%t Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Sep 01 2018, after _Seiichi Manyama_ *)
%Y Even-numbered terms of A005158.
%Y Cf. A005130, A006366, A049503.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 04 2001
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